Structures Informatiques et Logiques pour la Mod elisation - - PowerPoint PPT Presentation

Structures Informatiques et Logiques pour la Mod´ elisation Linguistique (MPRI 2.27.1 - second part)

Philippe de Groote

Inria

2015-2016

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 1 / 42

Semantic representations

1

Introduction

2

Modal logic Intension and extension Possibility and necessity Kripke semantics Hybrid Logic

3

Higher-order logic Simply typed λ-calculus Church’s simple theory of types Standard model Inherent incompleteness Henkin Models

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 2 / 42

Introduction

Semantic representations

1

Introduction

2

Modal logic Intension and extension Possibility and necessity Kripke semantics Hybrid Logic

3

Higher-order logic Simply typed λ-calculus Church’s simple theory of types Standard model Inherent incompleteness Henkin Models

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 3 / 42

Introduction

Semantics

Semantics is the study of meaning. In this setting, the logical meaning of a declarative utterance is reduced to its truth conditions (truth conditional semantics). Model-theoretic semantics: the logical meaning of a declarative utterance is captured by the set of models that make valid the interpretation of this utterance. Proof-theoretic semantics: the logical meaning of a declarative utterance is captured by a logical formula.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 4 / 42

Introduction

Example

John eats a red apple. ∃x.apple(x) ∧ red(x) ∧ eat(j, x)

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 5 / 42

Modal logic

Semantic representations

1

Introduction

2

Modal logic Intension and extension Possibility and necessity Kripke semantics Hybrid Logic

3

Higher-order logic Simply typed λ-calculus Church’s simple theory of types Standard model Inherent incompleteness Henkin Models

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 6 / 42

Modal logic Intension and extension

Semantic representations

1

Introduction

2

Modal logic Intension and extension Possibility and necessity Kripke semantics Hybrid Logic

3

Higher-order logic Simply typed λ-calculus Church’s simple theory of types Standard model Inherent incompleteness Henkin Models

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 7 / 42

Modal logic Intension and extension

Sinn und bedeutung

F.L.G. Frege (1848–1925)

Sinn (sense)/Bedeutung (reference) — Frege Intension/Extension — Carnap According to Frege, the sense of an expression is its “mode of presentation”, while the reference or deno- tation of an expression is the object it refers to. For instance, both expressions “1 + 1” and “2” have the same denotation but not the same sense.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 8 / 42

Modal logic Intension and extension

Intensional proposition

An intensional proposition is a proposition whose validity is not invariant under extensional substitution. Frege gives the example of “the morning star” and “the evening star” which both refer to the planet Venus. Compare “the morning star is the evening star” with “John does not know that the morning star is the evening star”.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 9 / 42

Modal logic Possibility and necessity

Semantic representations

1

Introduction

2

Modal logic Intension and extension Possibility and necessity Kripke semantics Hybrid Logic

3

Higher-order logic Simply typed λ-calculus Church’s simple theory of types Standard model Inherent incompleteness Henkin Models

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 10 / 42

Modal logic Possibility and necessity

Modals

In a strict sense, modal logic is concerned with the study of statements and reasonings that involve the notions of necessity and possiblity In a more general sense, modal logic is concerned with the study of statements and reasonings that involve expressions (modals) that qualify the validity of a judgement:

Alethic logic: It is necessary that... It is possible that... Deontic logic: It is mandatory that... It is allowed that... Epistemic logic: Bob knows that... Bob ignores that... Temporal logic: It will always be the case that... It will eventually be the case that...

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 11 / 42

Modal logic Possibility and necessity

Leibniz

G.W. von Leibniz (1646–1716)

A proposition is necessarily true if it is true in all possible worlds. A proposition is possibly true if it is true in at least one possible world. Pangloss enseignait la m´ etaphysico-th´ eologo-cosmolo-nigologie. Il prouvait admirablement qu’il n’y a point d’effet sans cause, et que, dans ce meilleur des mondes possibles, le chˆ ateau de monseigneur le baron ´ etait le plus beau des chˆ ateaux et madame la meilleure des baronnes possibles.

Voltaire (Candide)

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 12 / 42

Modal logic Possibility and necessity

Formalization

Syntax: F ::= a | ¬F | F ∨ F | F Define the other connectives in the usual way. Define ♦A as ¬¬A. A stands for “necessarily A”. ♦A stands for “possibly A”. Validity: let M = W, P, where W is a set of “possible worlds”, and P is a function that asigns to each atomic proposition a subset of W. M, s | = a iff s ∈ P(a). M, s | = ¬A iff not M, s | = A. M, s | = A ∨ B iff either M, s | = A or M, s | = B, or both. M, s | = A iff for every t ∈ W, M, t | = A.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 13 / 42

Modal logic Possibility and necessity

System S5

(P) all propositional tautologies (K) (A ⊃ B) ⊃ (A ⊃ B) (T) A ⊃ A (5) ♦A ⊃ ♦A Modus ponens: A ⊃ B A B Rule of necessitation: A A

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 14 / 42

Modal logic Kripke semantics

Semantic representations

1

Introduction

2

Modal logic Intension and extension Possibility and necessity Kripke semantics Hybrid Logic

3

Higher-order logic Simply typed λ-calculus Church’s simple theory of types Standard model Inherent incompleteness Henkin Models

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 15 / 42

Modal logic Kripke semantics

Kripke Semantics

let M = W, R, P, where W is a set of “possible worlds”, R is a binary relation over W, and P is a function that asigns to each atomic proposition a subset of W. M, s | = A iff for every t ∈ W such that sRt, M, t | = A. M, s | = ♦A iff for some t ∈ W such that sRt, M, t | = A.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 16 / 42

Modal logic Kripke semantics

System K

(P) all propositional tautologies (K) (A ⊃ B) ⊃ (A ⊃ B) Modus ponens: A ⊃ B A B Rule of necessitation: A A

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 17 / 42

Modal logic Kripke semantics

The following theorems of S5 are not valid in the class of all Kripke models: (D) A ⊃ ♦A (T) A ⊃ A (B) A ⊃ ♦A (4) A ⊃ A (5) ♦A ⊃ ♦A A binary relation R ⊂ W × W is serial if and only if for every s ∈ W there exists t ∈ W such that sRt.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 18 / 42

Modal logic Kripke semantics

Some well-known systems

KD basic deontic logic serial KT basic alethic logic reflexive KTB Brouwersche system reflexive, symmetric KT4 Lewis’ S4 reflexive, transitive KT5 Lewis’ S5 reflexive, symmetric, transitive

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 19 / 42

Modal logic Hybrid Logic

Semantic representations

1

Introduction

2

Modal logic Intension and extension Possibility and necessity Kripke semantics Hybrid Logic

3

Higher-order logic Simply typed λ-calculus Church’s simple theory of types Standard model Inherent incompleteness Henkin Models

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 20 / 42

Modal logic Hybrid Logic

Syntax

Key idea: provide the formula language with explicit means of speaking about worlds! Two sorts of atoms: usual atomic propositions (a, b, c, . . .), and nominals (i, j, k, . . .). Nominals will be used for naming worlds. F ::= a | i | ¬F | F ∨ F | F | ↓i. F | @iF ↓ is a binder: the free occurrences of i in A are bound in ↓i. F. On the,

• ther hand, @ is simply a binary connectives whose first term must be a

nominal. Intuition: ↓ is used for naming the “here-and-now”. It allows a nominal to be bound to the current world. @iA asserts that proposition A holds at world i.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 21 / 42

Modal logic Hybrid Logic

Semantics

Let M = W, R, P be a Kripke model, and let η be a valuation that assigns to each nominal an element of W. M, η, s | = a iff s ∈ P(a). M, η, s | = i iff s = η(i). M, η, s | = ¬A iff not M, η, s | = A. M, η, s | = A ∨ B iff either M, η, s | = A or M, η, s | = B, or both. M, η, s | = A iff for every t ∈ W such that sRt, M, η, t | = A. M, η, s | = ↓i. A iff M, η[i:=s], s | = A. M, η, s | = @iA iff M, η, η(i) | = A.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 22 / 42

Modal logic Hybrid Logic

Axiomatization

1

all propositional tautologies

2

↓i. (A ⊃ B) ⊃ (A ⊃ ↓i. B), where i does not occur free in A

3

↓i. A ⊃ (j ⊃ A[i:=j])

4

↓i. (i ⊃ A) ⊃ ↓i. A

5

↓i. A ≡ ¬↓i. ¬A

6

@i(A ⊃ B) ⊃ (@iA ⊃ @iB)

7

@iA ≡ ¬@i¬A

8

i ∧ A ⊃ @iA

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 23 / 42

Modal logic Hybrid Logic 9

@ii

10

@ij ⊃ (@jA ⊃ @iA)

11

@ij ≡ @ji

12

@i@jA ≡ @jA

13

♦@iA ⊃ @iA

14

♦i ∧ @iA ⊃ ♦A A ⊃ B A B A A A ↓ i. A A @iA @i(j ∧ A) ⊃ B (*) @iA ⊃ B @i♦(j ∧ A) ⊃ B (*) @i♦A ⊃ B (*) j is distinct from i and does not occur free in A or B.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 24 / 42

Modal logic Hybrid Logic

Example

The binary operator of temporal logic: A until B may be defined as: ↓i. ♦↓j.@i(♦(j ∧ B) ∧ (♦j ⊃ A))

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 25 / 42

Higher-order logic

Semantic representations

1

Introduction

2

Modal logic Intension and extension Possibility and necessity Kripke semantics Hybrid Logic

3

Higher-order logic Simply typed λ-calculus Church’s simple theory of types Standard model Inherent incompleteness Henkin Models

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 26 / 42

Higher-order logic Simply typed λ-calculus

Semantic representations

1

Introduction

2

Modal logic Intension and extension Possibility and necessity Kripke semantics Hybrid Logic

3

Higher-order logic Simply typed λ-calculus Church’s simple theory of types Standard model Inherent incompleteness Henkin Models

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 27 / 42

Higher-order logic Simply typed λ-calculus

What is lambda-calculus?

An intensional theory of functions. A simple functional programming language. A theory of free- and bound-variables, of scope and substitution. The keystone of higher-order syntax and higher-order logic. The algebra of natural-deduction proofs.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 28 / 42

Higher-order logic Simply typed λ-calculus

Syntax

Terms: T ::= x | λx. T | (T T) λ is a binder: thre free occurrences of x in t are bound in λx. t. Warning: You should solve, once and for all, any problem you could have with the notions of free and bound occurrences of variables. Reduction rule: (λx. t) u →β t[x:=u] Church-Rosser Theorem: For all λ-terms t, u, and v such that: t → →β u and t → →β v there exists a λ-term w such that: u → →β w and v → →β w

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 29 / 42

Higher-order logic Simply typed λ-calculus

Typing rules

Γ, x : A − x : A x : A, Γ − t : B Γ − λx. t : A → B Γ − t : A → B Γ − u : A Γ − (t u) : B Strong-Normalisation Theorem: There is no infinite reduction sequence.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 30 / 42

Higher-order logic Church’s simple theory of types

Semantic representations

1

Introduction

2

Modal logic Intension and extension Possibility and necessity Kripke semantics Hybrid Logic

3

Higher-order logic Simply typed λ-calculus Church’s simple theory of types Standard model Inherent incompleteness Henkin Models

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 31 / 42

Higher-order logic Church’s simple theory of types

Syntax

• A. Church

(1903–1995) Two atomic types: ι, o Logical constants: ⊥ :

• ⊃

:

• → o → o

∀α : (α → o) → o (at each type α) ι is the type of individuals and o is the type of propositions. Formulas are defined to be well-typed λ-terms of type o. We write P ⊃ Q and ∀x. P for ⊃ P Q and ∀α (λx. P), respectively. Similarly for the other connectives (¬, ∧, ∨, ≡, ∃), which are defined in the usual way. t = u is defined as ∀P.P t ⊃ P u.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 32 / 42

Higher-order logic Church’s simple theory of types

Deductive system

Logical rules: Γ, A − A Γ, A − B Γ − A ⊃ B Γ − A ⊃ B Γ − A Γ − B Γ − A x of type α, x ∈ FV (Γ) Γ − ∀α (λxα. A) Γ − ∀α A B of type α Γ − A B Γ, ¬A − ⊥ Γ − A

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 33 / 42

Higher-order logic Church’s simple theory of types

Deductive system

Conversion rule: Γ − A where A =β B Γ − B Extensionality axioms: Γ − (∀αx.A x = B x) ⊃ (A = B) Γ − (A ≡ B) ⊃ (A = B)

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 34 / 42

Higher-order logic Standard model

Semantic representations

1

Introduction

2

Modal logic Intension and extension Possibility and necessity Kripke semantics Hybrid Logic

3

Higher-order logic Simply typed λ-calculus Church’s simple theory of types Standard model Inherent incompleteness Henkin Models

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 35 / 42

Higher-order logic Standard model

Interpretation of the types and the terms

M = (Da)a∈T , I Dι is given. Do = {0, 1} DA→B = DBDA [[c]]η = I(c) [[x]]η = η(x) [[λx. t]]η = a → [[t]]η[x:=a] [[t u]]η = [[t]]η([[u]]η) With the expected interpretations for the logical constants.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 36 / 42

Higher-order logic Standard model

Higher-order logic as a set theory

Sets as characteristic functions, i.e., sets of “elements” of type α as terms of type α → o. {x | P} as λx. P t ∈ A as A t

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 37 / 42

Higher-order logic Inherent incompleteness

Semantic representations

1

Introduction

2

Modal logic Intension and extension Possibility and necessity Kripke semantics Hybrid Logic

3

Higher-order logic Simply typed λ-calculus Church’s simple theory of types Standard model Inherent incompleteness Henkin Models

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 38 / 42

Higher-order logic Inherent incompleteness

The set of natural numbers

S

△

= (∀x. s x = 0) ∧ (∀xy. s x = s y ⊃ x = y) N

△

= λx. (∀R. R 0 ∧ (∀y. R y ⊃ R (s y)) ⊃ R x) The only model of S ∧ ∀x. N x is the set of natural numbers.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 39 / 42

Higher-order logic Inherent incompleteness

Incompleteness

Let φ be a formula of Peano’s Arithmetic, and define φN as follows: φN = φ, for φ an atomic formula, (¬φ)N = ¬ φN, (φ ∗ ψ)N = φN ∗ ψN, for ∗ ∈ {∧, ∨, ⊃, ≡}, (∀x. φ)N = ∀x.(N x ⊃ φN), (∃x. φ)N = ∃x.(N x ∧ φN). Let D be the conjunction of the universal closures of the defining equations for addition and multiplication, and let PA be S ∧ ∀x. N x ∧ D. Then, the formula PA ⊃ φN is valid if and only if φ is true in the standard model of Peano’s arithmetic. Corollary: incompleteness of higher-order logic.

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 40 / 42

Higher-order logic Henkin Models

Semantic representations

1

Introduction

2

Modal logic Intension and extension Possibility and necessity Kripke semantics Hybrid Logic

3

Higher-order logic Simply typed λ-calculus Church’s simple theory of types Standard model Inherent incompleteness Henkin Models

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 41 / 42

Higher-order logic Henkin Models

Relaxing the interpretation

M = (Da)a∈T , I Dι is given. Do = {0, 1} DA→B ⊂ DBDA [[c]]η = I(c) [[x]]η = η(x) [[λx. t]]η = a → [[t]]η[x:=a] [[t u]]η = [[t]]η([[u]]η) With domains that are rich enough to interpret λ-abstraction, equality, and the logical constants .

Philippe de Groote (Inria) MPRI 2.27.1 2015-2016 42 / 42